I've played around 5 games to completion and many more partial games in the past few months on immortal and standard settings (no events), Communitas or Oval maps. I have a few thoughts on some balance issues. I would like to hear if others agree.
Ceremonial Burial
This belief is way too strong. The culture it provides is enormous, and the faith makes it snowball. The more faith you earn, the more cities you convert, the more faith you get. Just converting your own cities and some CS will easily get you to a second or third Prophet, which allows you to convert a neighbour even if they have a religion. If your neighbours have no religion, or if you score either of the theology wonders, things get even faster. This is exacebated by the AI generally being unable to defend against prophets. CB then synergises extremely well with For the Grace of God, and decently with many other beliefs. You can compare the numbers on CB to Transcendent Thoughts: TT yields 2-3 times the value of CB per city, but only triggers per era. You can easily get quite a lot of GPs per era, and their rate keeps increasing into the mid-late game, at times reaching one GP every few turns. In other words, CB totally outclasses TT, and reaches high numbers of cities much more easily due to the big faith yields. Holy Law is similarly outclassed. TT and HL both feel good in appropriate games, it's just that CB in insane. Of course CB requires you to be playing a GP strat to be effective, but for any tradition game I would choose CB over any other belief 100% of the time. I had a number of games where I was completing the third policy tree when the AI was completing their second. I think the concept of the belief is fine, it just needs smaller numbers, possibly combined with removing era scaling or changing the faith yield to something else.
Authority
Authority is great and can help you all game as you grind through the AI's army, and has interesting economic gameplay with tribute and border expansion, but there are many games when it is very weak as an opener. It especially has problems when you have resources which require many techs/workers to use. As authority, you really want to get horsemen, or perhaps triremes or spearmen, ASAP to kill units and get tribute. This is quite constraining since authority gets no early science bonuses, which I believe is its basic problem. You can find ways around this depending on the start, but in many cases you would do far better with progress. There are quite a few civs which mitigate the problems of authority, by having a strong early UU or tribute bonus, or other things like Russia's border yields. However for many civs it's a very inconsistent opener. This low science problem continues to medieval in a lot of games, and I would find myself wishing I'd gone progress as I watch every AI pumping medieval units while I'm still 12 turns from Steel. The bonuses from Imperium and Dominance are good, but limited and circumstantial. Having a militaristic CS to bully helps immensely but they are not too common. My suggestion would be to give authority some more science in some way, perhaps combined with a small nerf to something else. Something like the Ottomans' siege workshop science when producing units would fit thematically, or perhaps simply a flat science bonus of 10 when producing units, perhaps scaling slightly with more policies. This would add a lot of flexibility.
Some more minor thoughts:
Apostolic Tradition
This used to be very powerful as it gave culture. Massive food in the capital is cool, but I would only bother with it as tradition since the capital wants to work so many specialists. And as tradition, well there is ceremonial burial. The faith bonus of 2 per follower per tech is tiny.
Co-operation (and "on growth" strats in general)
I've tried these a couple of times and could not get them to work well. I would like to hear others' thoughts - maybe I was doing it wrong.
Abode of Peace
Seems very weak. The faith/gold is tiny, and minimum influence bonuses seem, to me, entirely useless. The quest reward bonus is nice, but rather low compared to Statecraft and Austria
Menin Gate
The bonus to improvements is cool. However 2 culture and GAP on own unit killed is tiny. Bonuses when your own units die feel quite underwhelming as the player will generally not lose a lot of units at most stages of the game, and this bonus would be small even if it was per enemy unit killed.
AI and Unit Upgrades
I feel that the AI doesn't upgrade its units aggressively enough. I often mow down dozens of AI swordsmen/spearmen when they could be upgrading them for a much stronger defence. Perhaps the AI lacks gold, but since the AI gets arbitrary bonuses anyway I would be happy to see it get some amount of free unit upgrades per turn, if that is technically possible.
Ceremonial Burial
This belief is way too strong. The culture it provides is enormous, and the faith makes it snowball. The more faith you earn, the more cities you convert, the more faith you get. Just converting your own cities and some CS will easily get you to a second or third Prophet, which allows you to convert a neighbour even if they have a religion. If your neighbours have no religion, or if you score either of the theology wonders, things get even faster. This is exacebated by the AI generally being unable to defend against prophets. CB then synergises extremely well with For the Grace of God, and decently with many other beliefs. You can compare the numbers on CB to Transcendent Thoughts: TT yields 2-3 times the value of CB per city, but only triggers per era. You can easily get quite a lot of GPs per era, and their rate keeps increasing into the mid-late game, at times reaching one GP every few turns. In other words, CB totally outclasses TT, and reaches high numbers of cities much more easily due to the big faith yields. Holy Law is similarly outclassed. TT and HL both feel good in appropriate games, it's just that CB in insane. Of course CB requires you to be playing a GP strat to be effective, but for any tradition game I would choose CB over any other belief 100% of the time. I had a number of games where I was completing the third policy tree when the AI was completing their second. I think the concept of the belief is fine, it just needs smaller numbers, possibly combined with removing era scaling or changing the faith yield to something else.
Authority
Authority is great and can help you all game as you grind through the AI's army, and has interesting economic gameplay with tribute and border expansion, but there are many games when it is very weak as an opener. It especially has problems when you have resources which require many techs/workers to use. As authority, you really want to get horsemen, or perhaps triremes or spearmen, ASAP to kill units and get tribute. This is quite constraining since authority gets no early science bonuses, which I believe is its basic problem. You can find ways around this depending on the start, but in many cases you would do far better with progress. There are quite a few civs which mitigate the problems of authority, by having a strong early UU or tribute bonus, or other things like Russia's border yields. However for many civs it's a very inconsistent opener. This low science problem continues to medieval in a lot of games, and I would find myself wishing I'd gone progress as I watch every AI pumping medieval units while I'm still 12 turns from Steel. The bonuses from Imperium and Dominance are good, but limited and circumstantial. Having a militaristic CS to bully helps immensely but they are not too common. My suggestion would be to give authority some more science in some way, perhaps combined with a small nerf to something else. Something like the Ottomans' siege workshop science when producing units would fit thematically, or perhaps simply a flat science bonus of 10 when producing units, perhaps scaling slightly with more policies. This would add a lot of flexibility.
Some more minor thoughts:
Apostolic Tradition
This used to be very powerful as it gave culture. Massive food in the capital is cool, but I would only bother with it as tradition since the capital wants to work so many specialists. And as tradition, well there is ceremonial burial. The faith bonus of 2 per follower per tech is tiny.
Co-operation (and "on growth" strats in general)
I've tried these a couple of times and could not get them to work well. I would like to hear others' thoughts - maybe I was doing it wrong.
Abode of Peace
Seems very weak. The faith/gold is tiny, and minimum influence bonuses seem, to me, entirely useless. The quest reward bonus is nice, but rather low compared to Statecraft and Austria
Menin Gate
The bonus to improvements is cool. However 2 culture and GAP on own unit killed is tiny. Bonuses when your own units die feel quite underwhelming as the player will generally not lose a lot of units at most stages of the game, and this bonus would be small even if it was per enemy unit killed.
AI and Unit Upgrades
I feel that the AI doesn't upgrade its units aggressively enough. I often mow down dozens of AI swordsmen/spearmen when they could be upgrading them for a much stronger defence. Perhaps the AI lacks gold, but since the AI gets arbitrary bonuses anyway I would be happy to see it get some amount of free unit upgrades per turn, if that is technically possible.